Water coming out from a fountain is modeled by the function f(x) = -x2 + 5x + 4 where f(x) represents the height, in feet, of the water from the fountain at different times x, in seconds.

What does the average rate of change of f(x) from x = 3 to x = 5 represent?

Question

Water coming out from a fountain is modeled by the function f(x) = -x2 + 5x + 4 where f(x) represents the height, in feet, of the water from the fountain at different times x, in seconds.

What does the average rate of change of f(x) from x = 3 to x = 5 represent?

ganeshie8 · Answer

rate of change of f(x) between x = a and x = b :  
$$\large \dfrac{f(b)-f(a)}{b-a}$$

ganeshie8 · Answer

a = 3
b = 5

evaluate f(3), f(5) and plugin

anonymous · Answer

I'm confused @ganeshie8

anonymous · Answer

So (5f - 3f)/5-3 ?

ganeshie8 · Answer

nice try, but no. you need to evaluate  :
$$\large \dfrac{f(5) - f(3)}{5-3}$$

ganeshie8 · Answer

f(5) is NOT same as 5f

f(5) is a notation which represents the function f(x) evaluated at x = 5

ganeshie8 · Answer

you have :
 f(x) = -x^2 + 5x + 4 

can you evaluate f(5) ?
f(5) = ?

anonymous · Answer

Ohhh I see now

anonymous · Answer

Give me an minute

anonymous · Answer

f(5) = 54

anonymous · Answer

f(3) = 28

anonymous · Answer

54 -28 / 5 - 3 = 13

ganeshie8 · Answer

careful, its -x^2

anonymous · Answer

But negative times a negative is positive right?

anonymous · Answer

So it wouldn't matter if the negative is there

ganeshie8 · Answer

f(x) = -x^2 + 5x + 4 

f(5) = -(5^2) + 5(5) + 4
      = -25     + 25   +  4
      =  0 + 4
      = 4

ganeshie8 · Answer

f(3) = -(3^2) + 5(3) + 4
       = -9  + 15  + 4
       = 10

ganeshie8 · Answer

the average rate would be :
$$\large  \dfrac{4 - 10}{5-3}$$

simplify

anonymous · Answer

-3

ganeshie8 · Answer

$\large \color{red}{\checkmark}$

ganeshie8 · Answer

Notice that by P `E`MDAS, Exponent should be evaluated before Multiplication, when you have an expression like   `-x^2`

ganeshie8 · Answer

-x^2  is not same as   (-x)^2

anonymous · Answer

So what does the average rate of change of f(x) from x = 3 to x = 5 represent?

anonymous · Answer

@ganeshie8

ganeshie8 · Answer

what do you think it represents ?

ganeshie8 · Answer

$$\large \text{Average rate of change}=\dfrac{f(b)-f(a)}{b-a}$$

anonymous · Answer

That the water comes out of the fountain at a speed of 3 feet per second

ganeshie8 · Answer

numerator has f(x) whose units are `feet` right?

ganeshie8 · Answer

and the denominator has units of seconds `seconds`

ganeshie8 · Answer

so the units of average rate of change would be  `feet/sec`
which is same as the units of speed, so yeah you're right !

anonymous · Answer

Awesome, thanks